Bone rotation behaves differently to object rotation
There are a few issues here:


The orientation of the bone at rest is not the same as the object. By default, the Y axis on each bone is along the main axis of the bone (i.e. the direction the bone is pointing). As blender uses Z-up as its standard coordinates, this most likely corresponds to the z axis on your object. 
Your object pivot point (the orangey dot when it is selected) does not match the pivot for the bone. This may, but not necessarily, cause differences in the rotations.
Euler rotations are not path independent, and as a result there are multiple ways to end up in the same orientation. Euler rotations are essentially a number of ordered rotations required to reach a given orientation, but there may be multiple ways to reach the same orientation. In this case try copying the values from the object to the bone and see which results you get.